Abstract
This paper considers a general stochastic SIR epidemic model driven by a multidimensional Lévy jump process with heavy-tailed increments and possible correlation between noise components. In this framework, we derive new sufficient conditions for disease extinction and persistence in the mean. Our method differs from previous approaches by the use of Kunita’s inequality instead of the Burkholder–Davis–Gundy inequality for continuous processes, and allows for the treatment of infinite Lévy measures by the definition of new threshold \(\mathcal {\bar{R}}_0\). An SIR model driven by a tempered stable process is presented as an example of application with the ability to model sudden disease outbreak, illustrated by numerical simulations. The results show that persistence and extinction are dependent not only on the variance of the processes increments, but also on the shapes of their distributions.
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Acknowledgements
This research is supported by the National Natural Science Foundation of China (No. 12001271) and Natural Science Foundation of Jiangsu Province (BK20200484), and by the Ministry of Education, Singapore, under its Tier 2 Grant MOE2016-T2-1-036.
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Communicated by Charles R. Doering.
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Privault, N., Wang, L. Stochastic SIR Lévy Jump Model with Heavy-Tailed Increments. J Nonlinear Sci 31, 15 (2021). https://doi.org/10.1007/s00332-020-09670-5
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DOI: https://doi.org/10.1007/s00332-020-09670-5
Keywords
- SIR model
- Multidimensional Lévy processes
- Extinction
- Persistence in the mean
- Kunita’s inequality
- Tempered stable process